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• http://dspace.wul.waseda.ac.jp/dspace/bitstream/2065/758/1/3603u145.pdf
平成 16 年度 修士論文

Newton-Cotes 求積法による数値積分の高精度保証

平成 17 年 2 月 2 日

指導教授：大石 進一 教授

早稲田大学大学院

理工学研究科

3603U145-2

情報ネットワーク専攻

森山 敦史• www.kashi.info.waseda.ac.jp/~yuichit/PDF/1g02p026.pdf

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• http://www.ccn.yamanashi.ac.jp/~stomo/pdf/nas2004.pdf

derogatory な行列の固有値問題
An eigenvalue problem for derogatory matrices
鈴木智博 （山梨大学），鈴木俊夫 （山梨大学）

1 はじめに一般的な固有値問題の解法に関しては，行列のサイズ，対称性，密度などに応じて様々な方法が存在し，利用可能な高性能なソフトウェアが存在する [3]．
行列 A ∈ Mn (K), K ∈ {R, C} の一つの固有値を λ とする．λ が代数的重複度 m(> 2) の重複固有値で，これに• http://www.doe.eng.cmu.ac.th/~sermsak/ee702/lnote3.pdf
付随する Jordan ブロックが 2 つ以上 m 未満あるとき，A を derogatory な行列と呼ぶ [7]．λ に付随する Jordan ブ
ロックが g (< m) 個存在するとき，この g は λ の幾何学的重複度と呼ばれ，λ に対応する g 個の固有ベクトルが存在する．

บทที่ 3

このような重複固有値の性質を知ることは，すなわち行列の Jordan 標準形を決定することである．F. Chatelinは [1] で，「Jordan 標準形は理論的には重要であるが数値的には安定でない」としている．また [6] では，固有ベクการวิเคราะห์จุดทำงานกระแสตรง

sio

いる．

n

トルの誤差限界を（精度保証付きで）見積もれるのは，その固有値の幾何学的重複度が 1 のときのみ，とされて今回，detogatory な行列の重複固有値，（一般）固有ベクトルを，そのレゾルベントに基づいて計算する一つの方การวิเคราะห์จุดทำงานกระแสตรง (DC Operating Point Analysis) เป็นการวิเคราะห์ ที่พิจารณา
ให้ค่าแรงดันและกระแสต่างๆ ในวงจรไม่มีการเปลี่ยนแปลงตามเวลา ดังนั้นอนุพันธ์เทียบกับเวลา
จึงเป็นศูนย์ เราจะได้สมการระบบเป็น

ver

法を報告する．

2 レゾルベントの級数展開

f (v) = i(v) + b(0) = GL v + iN (v) + b(0) = 0(3.1)• http://shop.rcd.ru/fulltext/215/1091การหาคำตอบ v ทำโดยใช้ วิธีการคำนวณเชิง เลข และต้องวนซ้ำ (iteration) ซึ่ง สรุป ขั้นตอน
A のレゾルベントの λ における Laurent 展開を考える．

lin

∞
∑
1. เลือกค่R(ζ)าเริ่มต้น
ดี v (0)ตัว=นับรอบ i = C
0. (ζ − λ)k=ที่ (A− และให้
ζI)−1
k
k=−∞
2. ถ้า kf (v (i) )k มีค่าน้อยมาก นั้นคือเราได้
คำตอบให้หยุดการทำงาน

on

(2.2)

e

ทั่วไปได้ดังนี้

(2.1)

1

∫

R(t)

Ck = ปรับแก้ pdt
3. คำนวณหาเวกเตอร์
2πi Γ (t − λ)k+1
4. ถ้า kpk มีค่าน้อยเกินไป นั้นคือเราอาจหยุดอยู่ที่ local minimum ให้หยุดการทำงาน

ただし，I は単位行列．また，積分路 Γ は固有値 λ をその内部に含みそれ以外の A の固有値を含まない，正の向きを持つ閉曲線とする．

5. v (i+1) = v (i) + αp, เมื่อ α เป็นค่าสเกลาร์ที่เหมาะสม

6. iC←2i +=
1 −C ．すなわち −C
レゾルベントの性質 [4] から，−1−1 は射影作用素である．これを P と表わす．また
−1

k−1
2
= −C−3 ，C−2 C−3 = −C7.
· ·maxIteration
から，D ให้
=หยุ−C
Cลั−k
= −D
C−2
า i· >
ดการทำงาน
มิเช่นนั้นให้ก
บไปทำขั
้นตอนที่ 2, k = 2, 3, . . . となる．さらに−4ถ้，−2 とすると，
S = C0 とすると Ck = S k+1 , k = 0,1,.
.
.
である．これらを用いて
(2.1)
は，
(i)

ในที่นี้ v(2.3)

แทนค่าของ v ในรอบการวนซ้ำที่ i

l−1
∞
∑
∑
Dk
P
R(ζ) = −(ζ − λ)k S k+1−+
3.1 วิธีนิวตัน (ζ − λ)k+1
ζ −λ
k=1

k=0

วิธีนิวตัน (Newton or Newton-Raphson) จะคำนวณค่าเวกเตอร์ปรับแก้ p จากความสัมพันธ์

と表わせる．P ，D，S をそれぞれ λ に付随するスペクトル射影，べき零行列，縮小レゾルベントと呼ぶ [1]．ま(i)∂f (v )た，λ は R の極であり，極の位数 l は C−l 6= 0 であるような最大の整数である．さらに，この
p = J (v (i) )p = −f (v (i) )(3.2) l は固有値 λ の指(i)∂v

標と呼ばれ，λ に付随する最大の Jordan ブロックのサイズに等しい．
21• http://yebisu.cc.kyushu-u.ac.jp/~watanabe/LECTURE/INA/04.pdf

情報数値解析
第４回

区間演算

2009 56
年 11 月 24 日